Theory

Transfer-Matrix Method

From Snell's law and Fresnel boundaries to multilayer matrix solving

This chapter presents TMM in the order of interface laws, in-layer phase propagation, and full-stack matrix composition, then maps those equations to app outputs.

Chapter Scope

Level	Physical statement	Simulation consequence
Refraction law	Angle transformation across interfaces	Internal propagation angles are determined for each layer
Fresnel boundary law	Amplitude split at each interface	Polarization-dependent reflection and transmission
In-layer phase evolution	Thickness-dependent phase accumulation	Spectral shift of resonances and fringes
Matrix chain	Ordered composition across all layers	Final (R), (T), (A), ellipsometry, and depth-resolved quantities

Snell's Law and Angular Refraction

For an interface between media (i) and (j), the propagation angles satisfy

n_i \sin \theta_i = n_j \sin \theta_j .

In multilayers, this relation is applied recursively at every boundary. Therefore, the single input Incident Angle determines the angular state throughout the stack.

$Snell's law refraction geometry$

Fresnel Equations and Interface Coefficients

Field continuity at the interface yields amplitude coefficients. For non-magnetic media, a standard angular form is

r_s = \frac{n_i \cos\theta_i - n_j \cos\theta_j}{n_i \cos\theta_i + n_j \cos\theta_j}, \qquad t_s = \frac{2 n_i \cos\theta_i}{n_i \cos\theta_i + n_j \cos\theta_j},

r_p = \frac{n_j \cos\theta_i - n_i \cos\theta_j}{n_j \cos\theta_i + n_i \cos\theta_j}, \qquad t_p = \frac{2 n_i \cos\theta_i}{n_j \cos\theta_i + n_i \cos\theta_j}.

Power coefficients are derived from amplitudes:

R = |r|^2,\qquad T = \Re\!\left(\frac{n_j \cos\theta_j}{n_i \cos\theta_i}\right)|t|^2,\qquad A = 1 - R - T .

Fresnel reflectance of s and p polarization versus incident angle

*Source: Wikimedia Commons, "Fresnel equations - reflectance.svg". Original file page: *

Phase Accumulation Inside a Layer

For a layer of thickness (d), refractive index (n), and internal angle (\theta), the phase thickness is

\delta = k_0 n d \cos\theta,\qquad k_0 = \frac{2\pi}{\lambda}.

This term governs the displacement of interference extrema under thickness or wavelength variation.

Transfer-Matrix Composition for Multilayers

The TMM formulation combines interface and propagation operators:

\mathbf{M}_{\text{total}} = \mathbf{M}_{01}\mathbf{P}_1\mathbf{M}_{12}\mathbf{P}_2\cdots\mathbf{M}_{N,N+1}.

A commonly used propagation matrix is

\mathbf{P}_\ell= \begin{bmatrix} e^{i\delta_\ell} & 0\\ 0 & e^{-i\delta_\ell} \end{bmatrix}.

The global matrix returns (r) and (t), then all power and phase observables follow from those amplitudes.

Mapping to App Outputs

App output	Physical origin	Interpretation focus
`Reflection`	`R = P_ref / P_inc` (reflected-to-incident power ratio)	Return-power fraction and stop-band behavior
`Transmission`	`T = P_tr / P_inc` (transmitted-to-incident power ratio with medium/angle scaling)	Through-power fraction and coupling efficiency
`Absorption`	`1 - R - T`	Net dissipative loss
`Layer Absorption`	Absorption decomposition by layer	Dominant loss layer localization
`Ellipsometry`	`rho = r_p / r_s = tan(Psi) * exp(iDelta)`	Polarization phase and amplitude contrast
`Depth Distribution`	Spatial field solution	Field maxima, nodes, and absorption hotspots

External References

Continue with Optical Concepts and Structure Configuration.

Quick Start

Build a DBR structure and complete your first calculation

Optical Concepts

Refractive index, polarization, angle, birefringence, and ellipsometry